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Unformatted text preview: the endpoints. Since f is bounded, there exists m,M R so that m f ( x ) M for all x [ a,b ]. Let > 0 be given. We set = / 2( Mm ). Since f is continuous on [ a,b ], there is a partition P of [ a,b ] so that U ( P,f )L ( P,f ) < / 2 . As a partition for [ a,b ], we set P = P { b } . Then, U ( P,f )L ( P,f ) = U ( P,f )L ( P,f ) + ( M *m * ) where M * ,m * are the sup and inf of f over [ b,b ] respectively. Then, U ( P,f )L ( P,f ) < ( / 2) + ( Mm ) = . Since > 0 was arbitrary, this shows that f R [ a,b ] as desired. 1...
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 Spring '10
 MetCalfe
 Mean Value Theorem

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